Optimization for Compressed Sensing: the Simplex Method and Kronecker Sparsification

TL;DR

This paper introduces two novel methods for large-scale compressed sensing: one using a simplex-based approach for sparse signals, and another employing Kronecker sensing to exploit matrix structures for faster solutions.

Contribution

The paper presents two independent approaches—simplex method variants for sparse signals and Kronecker sensing—that significantly improve computational efficiency in large-scale compressed sensing.

Findings

Simplex variants reduce computation time for very sparse signals.

Kronecker sensing enables faster solutions due to sparser problem formulation.

Numerical results show up to ten-fold speedup in solving compressed sensing problems.

Abstract

In this paper we present two new approaches to efficiently solve large-scale compressed sensing problems. These two ideas are independent of each other and can therefore be used either separately or together. We consider all possibilities. For the first approach, we note that the zero vector can be taken as the initial basic (infeasible) solution for the linear programming problem and therefore, if the true signal is very sparse, some variants of the simplex method can be expected to take only a small number of pivots to arrive at a solution. We implemented one such variant and demonstrate a dramatic improvement in computation time on very sparse signals. The second approach requires a redesigned sensing mechanism in which the vector signal is stacked into a matrix. This allows us to exploit the Kronecker compressed sensing (KCS) mechanism. We show that the Kronecker sensing requires stronger conditions for perfect recovery compared to the original vector problem. However, the Kronecker sensing, modeled correctly, is a much sparser linear optimization problem. Hence, algorithms that benefit from sparse problem representation, such as interior-point methods, can solve the Kronecker sensing problems much faster than the corresponding vector problem. In our numerical studies, we demonstrate a ten-fold improvement in the computation time.